The force per unit length existing between two infinite parallel conductors is given by ___________

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Examoogle · Accepted Answer

Option A: $\frac{\mu_0 I_1 I_2}{2\pi d}$

Explanation & Concepts:
The force per unit length existing between two parallel conductors carrying currents $I_1$ and $I_2$ separated by a distance $d$ in free space is given by $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$. This fundamental relationship follows from Ampere's Law and the Lorentz force principle, where the magnetic field produced by one conductor exerts a force on the current-carrying second conductor.

Working Principle: A current $I_1$ flowing through an infinitely long conductor produces a magnetic flux density $B_1 = \frac{\mu_0 I_1}{2\pi d}$ at a perpendicular distance $d$. When a second conductor carrying current $I_2$ is placed in this magnetic field, it experiences a magnetic force per unit length given by $F/L = B_1 I_2 = \frac{\mu_0 I_1 I_2}{2\pi d}$. If currents flow in the same direction, the force is attractive; if in opposite directions, it is repulsive.

Key Formulas: $F/L = \frac{\mu_0 I_1 I_2}{2\pi d}$ — Force per unit length between two long parallel conductors; $B = \frac{\mu_0 I}{2\pi d}$ — Magnetic flux density around a long straight current-carrying wire

Key Points: Parallel currents in the same direction attract each other.; Antiparallel (opposite direction) currents repel each other.; This formula is used to define the standard SI unit of current, the Ampere.; The force is directly proportional to the product of currents ($I_1 I_2$) and inversely proportional to distance ($d$).

Applications: Definition of the standard SI unit of current (Ampere).; Calculating electrodynamic forces on busbars in high-voltage switchgear during short-circuit conditions.

Additional Info: Option B lacks the $\pi$ term in the denominator, which is required due to the cylindrical symmetry of the magnetic field.. Option C lacks both the constant 2 and $\pi$ in the denominator.. Option D represents a expression missing the second current $I_2$ and required constants.

Answer: A is correct — The magnetic force per unit length between two infinite parallel current-carrying conductors is $\frac{\mu_0 I_1 I_2}{2\pi d}$.

Exam Tip: Remember that parallel conductors with currents in the SAME direction ATTRACT, which is opposite to electrostatic charges where LIKE charges REPEL.

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Option B: $\frac{\mu_0 I_1 I_2}{2d}$

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Option C: $\frac{\mu_0 I_1 I_2}{d}$

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Option D: $\frac{\mu_0 I_1}{d}$

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